Introduction

xxi

as a sum of four squares by replacing squares with triangular numbers,

obtaining, in our view, a cleaner result. A last example of the type of result

we will discuss in this chapter is the following. Let S be a set consisting

of the positive integers with an additional copy of those positive integers

congruent to zero modulo 7. Decompose S into its even and odd parts, E

and O respectively. Denote by PE{^) and Po(k) the number of partitions

of k with parts taken from the sets E and O respectively. We prove that for

all nonnegative integers fc,

PE(2k) = P0(2k + l).

The prerequisites for this book are a thorough understanding of material

traditionally covered in first year graduate courses - especially the contents

of the complex analysis course. We review, however, the most salient points

about elliptic function theory portions of this course. Although a knowledge

of Riemann surfaces and Fuchsian groups is helpful, it is not needed by the

reader who is willing to accept the summaries of the required material (with

references to the literature). Although we do not, in general, reproduce

material available in other textbooks, we make an exception for material

on theta functions and theta constants despite the availability of excellent

sources (for example, [23]). We do so, not only for the convenience of the

reader, but also to emphasize our point of view. We have also ignored, to a

great extent, the combinatorial and special functions connection. These are

discussed in [2], [7] and [10], for example.

A road map

While we did not intend to write an encyclopedic text, the result has been

quite a large book. We take the liberty of offering the readers our suggestions

for possible ways of going through this text, which was written with several

different types of readers in mind. These range from the beginning grad-

uate mathematics student through the professional mathematician whose

interests are either in combinatorial mathematics (partition theory, repre-

sentation as sums of squares, counting points on conic sections) or function

theory (Riemann surfaces, modular forms). Theoretical physicists might be

interested in portions of the material we cover.

The reader is expected to have a reasonable knowledge of the theory of

functions of a complex variable, through the Riemann mapping theorem,

and enough mathematical maturity to follow an argument even though un-

familiar with the proofs of all the tools used. Thus, the book can be used as

a text for a topics course in either analysis or analytic number theory, and as